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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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Steady-State Errors

• Three performance criteria in analysis and design
of control systems:
o Transient response
o Stability
o Steady-state errors

• Transient response of 1st and 2nd order systems
have been discussed in previous lectures.

• This section focuses on steady-state errors of the
time response of a particular system.

Definition

• Steady-state error is the difference between the

input and the output for a prescribed test input as
t → ∞.
• Test inputs used for steady-state error analysis and
design are summarized in Table 7.1.

• Use for stable systems only.

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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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Table 7.1

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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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Steady-state errors:

• Output 1: zero ess; Output2: ess = e2 (∞)

• Output 1: zero ess; Output2: ess = e2 (∞)

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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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• Steady-state error, ess can be obtained using the
final value theorem:

• Example: Find the steady-state error for an open
5
and the input
loop system with G ( s ) = 2

s + 7 s + 10
is unit step input.

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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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Steady-state errors for Unity Feedback
Systems
• Consider a unity feedback system

• Assume system is stable, we can apply the final
value theorem:

• The steady-state error depends on the input signal.

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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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Step input
1
• Consider the case when R ( s ) = ,
s

• Hence in order to have zero steady state error, the
term, lim G ( s ) , must be big. i.e.:
s→ 0

• This can only happen if

• If n = 0 ,

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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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Ramp Input
• Consider the case when r (t ) = t , R ( s ) =

1
.
2
s

• Hence, to have zero steady state error for a ramp
input,

• This can only happen if

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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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Parabolic input
1
1
• Consider the case when r (t ) = t 2 , R ( s ) = 3 .
2
s
• Hence, to have zero steady state error for a ramp
input,

• This can only happen if

• If n=2,

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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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• Example: Find the steady-state error for input
u (t ), tu (t ), t 2 u (t ) to the system shown below. The
function u (t ) is the unit step.

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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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• Example: Find the steady-state error for input
5u (t ), 5tu (t ), 5t 2 u (t ) to the system shown below.
The function u (t ) is the unit step.

ZHI